There is a widely known intriguing phenomenon that discrete-time GARCH and stochastic volatility (SV) models share the same continuous-time diffusion model as their weak convergence limit, but statistically, the GARCH model is not asymptotically equivalent to the SV or diffusion model. This paper investigates GARCH-type quasi-likelihood ratios for the SV and diffusion models whose own likelihoods are analytically intractable. We show that the two quasi-likelihood ratios for the SV and diffusion models asymptotically have the same closed-form expression that is different from the limiting likelihood ratio of the GARCH model.

中文翻译：

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SV 模型的 GARCH 准似然比和扩散极限

有一个广为人知的有趣现象，即离散时间 GARCH 和随机波动率 (SV) 模型共享相同的连续时间扩散模型作为它们的弱收敛极限，但在统计上，GARCH 模型并不渐近等效于 SV 或扩散模型。本文研究了 SV 和扩散模型的 GARCH 型准似然比，其自身的似然性在分析上难以处理。我们表明 SV 和扩散模型的两个拟似然比渐近地具有相同的封闭形式表达式，这与 GARCH 模型的极限似然比不同。